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True Infinitesimal Differential Geometry Mikhail G. Katz∗ Vladimir

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True infinitesimal differential geometry Mikhail G. Katz∗ Vladimir Kanovei Tahl Nowik M. Katz, Department of Mathematics, Bar Ilan Univer- sity, Ramat Gan 52900 Israel E-mail address: [email protected] V. Kanovei, IPPI, Moscow, and MIIT, Moscow, Russia E-mail address: [email protected] T. Nowik, Department of Mathematics, Bar Ilan Uni- versity, Ramat Gan 52900 Israel E-mail address: [email protected] ∗Supported by the Israel Science Foundation grant 1517/12. Abstract. This text is based on two courses taught at Bar Ilan University. One is the undergraduate course 89132 taught to a group of about 120 freshmen. This course used infinitesimals to introduce all the basic concepts of the calculus, mainly following Keisler’s textbook. The other is the graduate course 88826 taught to about 10 graduate students. The graduate course developed two viewpoints on infinitesimal generators of flows on manifolds. Much of what we say here is deeply affected by the pedagogical experience of teaching the two groups. The classical notion of an infinitesimal generator of a flow on a manifold is a (classical) vector field whose relation to the actual flow is expressed via integration. A true infinitesimal framework allows one to re-develop the foundations of differential geometry in this area. The True Infinitesimal Differential Geometry (TIDG) framework enables a more transparent relation between the infini- tesimal generator and the flow. Namely, we work with a combinatorial object called a hyper- real walk constructed by hyperfinite iteration. We then deduce the continuous flow as the real shadow of the said walk. Namely, a vector field is defined via an infinitesimal displace- ment in the manifold itself. Then the walk is obtained by iteration rather than integration. We introduce synthetic combinatorial con- ditions Dk for the regularity of a vector field, replacing the classical analytic conditions of Ck type, that guarantee the usual theorems such as uniqueness and existence of solution of ODE locally, Frobe- nius theorem, Lie bracket, small oscillations of the pendulum, and other concepts. Here we cover vector fields, infinitesimal generator of flow, Frobenius theorem, hyperreals, infinitesimals, transfer principle, ultrafilters, ultrapower construction, hyperfinite partitions, micro- continuity, internal sets, halo, prevector, prevector field, regularity conditions for prevector fields, flow of a prevector field, relation to classical flow. Contents Chapter 1. Preface 11 Part 1. True Infinitesimal Differential Geometry 13 Chapter 2. Introduction 15 2.1. Breakdown by chapters 16 2.2. Historical remarks 17 2.3. Preview of flows and infinitesimal generators 18 Chapter 3. A hyperreal view 19 3.1. Successive extensions N, Z, Q, R, ∗R 19 3.2. Motivating discussion for infinitesimals 20 3.3. Introduction to the transfer principle 21 3.4. Transfer principle 24 3.5. Orders of magnitude 25 3.6. Standard part principle 27 3.7. Differentiation 27 Chapter 4. Ultrapower construction and transfer principle 31 4.1. Introduction to filters 31 4.2. Examples of Filters 32 4.3. Properties of filters 33 4.4. Real number field as quotient of sequences of rationals 34 4.5. Extension of Q 34 4.6. Examples of infinitesimals 34 4.7. Ultrapower construction of the hyperreal field 35 Chapter 5. Microcontinuity, EVT, internal sets 37 5.1. Construction via equivalence relation 37 5.2. Extending the ordered field language 37 5.3. Existential and Universal Transfer 39 5.4. Examples of first order statements 39 5.5. Galaxies 41 5.6. Microcontinuity 43 5.7. Uniform continuity in terms of microcontinuity 45 3 4 CONTENTS 5.8. Hyperreal extreme value theorem 46 5.9. Intermediate value theorem 47 5.10. Ultrapower construction applied to (R); internal sets 48 5.11. The set of infinite hypernaturals is notP internal 49 5.12. Transfering the well-ordering of N; Henkin semantics 50 5.13. From hyperrationals to reals via ultrapower 51 5.14. Repeatingtheconstruction 53 Chapter 6. Application: small oscillations of a pendulum 55 6.1. Small oscillations of a pendulum 55 6.2. Vector fields, walks, and flows 56 6.3. The hyperreal walk and the real flow 57 6.4. Adequality pq 58 6.5. Infinitesimal oscillations 59 6.6. Linearisation 59 6.7. Adjusting linear prevector field 60 6.8. Conclusion 60 Chapter 7. Differentiable manifolds 63 7.1. Definition of differentiable manifold 63 7.2. Open submanifolds, Cartesian products 64 7.3. Tori 65 7.4. Projective spaces 66 7.5. Definition of a vector field 67 7.6. Zeros of vector fields 68 7.7. 1-parameter groups of transformations of a manifold 69 7.8. Invariance of infinitesimal generator under flow 70 7.9. Lie derivative and Lie bracket 71 Chapter 8. Theorem of Frobenius 73 8.1. Invariance under diffeomorphism 73 8.2. Frobenius thm, commutation of vector fields and flows 75 8.3. Proof of Frobenius 76 8.4. A-trackandB-trackapproaches 78 8.5. Relations of and 78 8.6. Smooth manifolds;≺ notion≺≺ nearstandardness 79 8.7. Results for general culture on topology 80 Chapter 9. Gradients, Taylor, prevectorfields 83 9.1. Propertiesofthenaturalextension 83 9.2. Remarks on gradient 83 9.3. Prevectors 85 9.4. Tangent space to a manifold via prevectors 86 CONTENTS 5 9.5. Action of a prevector on smooth functions 87 9.6. Maps acting on prevectors 88 Chapter 10. Prevector fields, classes D1 and D2 91 10.1. Prevector fields 91 10.2. Realizing classical vector fields 92 10.3. Regularity class D1 for prevectorfields 94 10.4. Second differences, class D2 94 10.5. Secord-order mean value theorem 95 10.6. The condition C2 implies D2 96 10.7. Bounds for D1 prevector fields 97 Chapter 11. Prevectorfield bounds via underspill, walks, flows 99 11.1. Operations on prevector fields 99 11.2. Sharpening the bounds via underspill 100 11.3. Further bounds via underspill 101 11.4. Global prevector fields, walks, and flows 102 11.5. Internal induction 104 11.6. Dependence on initial condition and prevector field 104 11.7. Dependence on initial conditions for local prevector field 106 11.8. Inducing a real flow 108 11.9. Geodesic flow 109 Chapter 12. Universes, transfer 111 12.1. Universes 111 12.2. Therealsasthebaseset 112 12.3. More sets in universes 113 12.4. Membership relation 114 12.5. Ultrapower construction of nonstandard universes 114 12.6. Los theorem 117 12.7. Elementary embedding 119 12.8. Definability 122 12.9. Conservativity 123 Chapter13. Moreregularity 127 13.1. D2 implies D1 127 13.2. Injectivity 128 13.3. Time reversal 129 Part 2. 500 yrs of infinitesimal math: Stevin to Nelson 131 Chapter 14. 16th century 133 14.1. Simon Stevin 133 14.2. Decimals from Stevin to Newton and Euler 135 6 CONTENTS 14.3. Stevin’s cubic and the IVT 136 Chapter 15. 17th century 139 15.1. Pierre de Fermat 139 15.2. James Gregory 139 15.3. Gottfried Wilhelm von Leibniz 139 Chapter 16. 18th century 141 16.1. Leonhard Euler 141 Chapter 17. 19th century 143 17.1. Augustin-Louis Cauchy 143 Chapter 18. 20th century 145 18.1. Thoralf Skolem 145 18.2. Edwin Hewitt 145 18.3. JerzyLo´s 145 18.4.AbrahamRobinson 145 18.5. Edward Nelson 145 Bibliography 147 Chapter 19. Theorema Egregium 153 19.1. Preliminaries to the Theorema Egregium of Gauss 153 19.2. The theorema egregium ofGauss 155 19.3. Intrinsic versus extrinsic 156 19.4. Gaussian curvature and Laplace-Beltrami operator 157 19.5. Duality in linear algebra 158 19.6. Polar, cylindrical, and spherical coordinates 159 19.7. Duality in calculus; derivations 161 Chapter20. Derivations,tori 163 20.1. Space of derivations 163 20.2. Constructing bilinear forms out of 1-forms 165 20.3. Firstfundamentalform 165 20.4. Expressing the metric in terms of 1-forms 166 20.5. Hyperbolic metric 167 20.6. Surface of revolution 167 20.7. Coordinate change 169 20.8. Conformal equivalence 170 20.9. Uniformisation theorem for tori 170 20.10. Isothermalcoordinates 171 20.11. Tori of revolution 172 Chapter 21. Conformal parameter 175 CONTENTS 7 21.1. Conformal parameter τ of standard tori of revolution 175 21.2. Comformal parameter ofstandard torus ofrev 176 21.3. Curvature expressed in terms of θ(s) 178 21.4. Total curvature of a convex Jordan curve 179 ˜ 21.5. Signed curvature kα of Jordan curves in the plane 180 21.6. Coordinate patch defined by family of normal vectors 183 21.7. Total Gaussian curvature and Gauss–Bonnet theorem 185 Chapter 22. Addendum on a Bernoullian framework 187 22.1. Ultrapower construction 187 22.2. Saturation and topology 187 Chapter 23. More on Gauss-Bonnet 189 23.1. Second proof of Gauss–Bonnet theorem 189 23.2. Euler characteristic 189 23.3. Exercise on index notation 190 Chapter 24. Bundles 191 24.1. Trivial bundle (E,B,π), function as a section (mikta) 191 24.2. Mobius band as a first example of a nontrivial bundle 192 24.3. Tangent bundle 192 24.4. Hairy ball theorem 193 24.5. Cotangent bundle 193 24.6. Exterior derivative of a function 194 24.7. Extending gradient and exterior derivative 194 24.8. Plan for building de Rham cohomology 195 24.9. Exterioralgebra,area,determinant,rank 196 24.10. Formal definition, alternating property 196 24.11. Example: rank 1 197 24.12. Areas in the plane 197 24.13. Vectorandtripleproducts 199 24.14. Anticommutativity of the wedge product 200 24.15. The k-exterior power; simple (decomposable) multivectors200 24.16. Basis and dimension of exterior algebra 201 24.17. Rank of a multivector 201 Chapter 25. Multilinear forms; exterior differential complex 203 25.1.
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